Coster and Cheng presented a generalized foldover scheme for the construction of systematic run orders of fractional factorial designs, with all factors having the same prime power number of levels, for which all the main effects components of the factors are orthogonal to a polynomial trend present in every block of the design. In this paper, we present modifications to the foldover method that allow polynomial trend-free run orders to be constructed in the following more general settings: Designs for which the number of levels of each factor is not a prime power; mixed-level factorial designs with factors at different numbers of levels; cases in which some or all two- and higher-factor interactions, not just the main effects, are required to be orthogonal to the polynomial trend.